In this paper we consider a class of strongly singular, nonlinear integral equations. A numerical method is proposed and its convergence is proved in weighted Sobolev spaces.

Collocation and quadrature methods for singular integro-differential equations of Prandtl's type are studied in weighted Sobolev spaces as well as in weighted spaces of continuous functions. A fast algorithm based on the quadrature method is proposed. Convergence results and error estimates are given.

We deal with the numerical evaluation of integrals of functions with strong singularities often used in applications. The convergence and the stability of the methods for the numerical computation of such integrals is investigated. The goodness of the numerical results for pratical applications are examined.

Convergence and boundedness of the extended Lagrange interpolating operator with additional nodes are studied in the space L_p^{u,t } of Sobolev type.

An algorithm for the approximate evaluation of the Hilbert transform has been prposed. The convergence of the procedure is proved. The stability of the algorithm is considered and some numerical examples are given.

We consider a class of integral equations of Volterra type with constant coefficients containing a logarithmic difference kernel. This class coincides for a=0 with the Symm's equation. We can transform the general integral equation into an equivalent singular equation of Cauchy type which allows us to give an explicit formula for the solution g. The numerical method proposed in this paper consists in substituting the Lagrange polynomial interpolating the known function f in the expression of the solution g. Then, with the aid of the invariance properties of the orthogonal polynomials for the Cauchy integral equations, we obtain an easy expression for the approximate solution. Moreover, we show that the previous numerical method is a collocation method where the coefficient of the polynomial approximating the solution can be easily computed. We give weighted norm estimates for the error of this method. The paper concludes with some numerical examples.

We consider a class of integral equations of Volterra type with constant coefficients containing a logarithmic difference kernel. This class coincides for a=0 with the Symm's euqtion. We can transform the general integral equation into an equivalent singular equation of Cauchy type which allows us to give the explicit formula for the solution. The numerical method proposed in this paper consists in substituting this in the experrsion of the solution g. Then, with the aid of the inveriance properties of the orthogonal polynomials for the Cauchy integral equation, we obtain an approximate solution of the function g. We give weighted norm estimates for the error of this method. The paper concludes with some numerical examples.

The authors introduce a new Lagrange and Hermite interpolation process and study the behaviour in some functional spaces.

Collocation and quadrature methods for singular integro-differential equations of Prandtl's type are studied in weighted Sobolev spaces. A fast algorithm basing on the quadrature method is proposed. Convergence results and error estimates are given.

The authors show the uniform boundedness of the Lagrange operator in some weighted Sobolev-type space.

The authors study the convergence and the stability of a collocation and a discrete collocation method for Cauchy singular integral equations with weakly singular perturbation kernels in some weighted uniform norms. Uniform error estimates are also given. © 1997 Rocky Mountain Mathematics Consortium.

The Hilbert transform of a function g, H(g) is an important tool in many mathematical fields. Expecially its numerical evaluation is often useful in some procedures for searcing solutions of the singular integral equations. In this context an approximation of (HV^alpha,beta,f;t), |t|1, where f is a continuous function in [-1,1] and v^alpha,beta, alpha,beta>-1 is a Jacobi weight, is required. In the last decade more then one paper appeared on this subject and among others we recall [1,2,3,4,5,14,15,20]. The procedure used in these papers can be described as follows. Approximate the function f with a Lagrange interpolating polynomial and obtain a quadrature formula with coefficients depending on the singularity t. The error of the quadrature formula was estimated assuming t in a "more or less" wide neighborhood of zero since (HV^alpha,beta,f;t) is unbounded at the endpoints of (-1,1). In this paper we propose a more accurate procedure: since (HV^alpha,beta,f;t) is unbounded at the endpoints, itis more natural to consider its approximation in some functional spaces equipped with weighted uniform norms. In Section 2 (see theorems 2.1 and 2.2) we state the behaviour of (HV^alpha,beta,f;t) in [-1,1] and we give new conditions for its exstence. In sections 3 and 4 we cosntruct approximations of (HV^alpha,beta,f;t) which converge to the exact value in weighted uniform norm. The L^p convergence of the formula is also briefly treated. Finally since the considered procedures are numerically stable, these can be used for the numerical evaluation of (HV^alpha,beta,f;t) . In this sense the present paper is a short survey on the subject.

Alcuni modelli di dispersione di sostanze inquinanti prodotte dal traffico autoveicolare in un area urbana vengono presentati. In particolare si descrive il modello OMG-Volume Source che sembra essere in grado di analizzare realisticamente l'effetto "canyon", tipico di un area urbana. I parametri del modello vengono determinati tenendo soprattutto in considerazione le caratteristiche spaziali dell'insediamento urbano.

Alcuni modelli di dispersione di sostanze inquinanti prodotte dal traffico autoveicolare in un area urbana vengono presentati. In particolare si descrive il modello OMG-Volume Source che sembra essere in grado di analizzare realisticamente l'effetto "canyon", tipico di un area urbana. I parametri del modello vengono determinati tenendo soprattutto in considerazione le caratteristiche spaziali dell'insediamento urbano.

Collocation and quadrature methods for Cauchy singular integral equations on an interval with variable coefficients are studied. Convergence rates are proved in weighted uniform and uniform norms.

We consider a class of integral equations of Volterra type with constant coefficients containing a logarithmic difference kernel. This equation can be transformed into an equivalent singular equation of Cauchy type which allows us to give the explicit formula for the solution. The numerical method proposed in this paper consists of applying the Lagrange interpolation to the inner Cauchy type singular integral in the latter formula after subtracting the singularity. For the error of this method weighted norm estimates as well as estimates on discrete subsets of knots are given. The paper concludes with some numerical examples. © 1995 Rocky Mountain Mathematics Consortium.

Boundedness and convergence of the extended Lagrange interpolating operator are investigated in the space L_u,t^p of Sobolev type.

The authors study the convergence and the stability of a collocation and a discrete collocation method for Cauchy singular integral equations with weakly singular perturbation kernel in some weighted uniform norms. Uniform error estimates are also given.

Using the simulated annealing technique we re-examine the role played by the minimization of the physicochemical distances between amino acids during the origin of the organization of the genetic code. The results are discussed in the context of the various hypotheses proposed to explain how amino acids were allocated in the genetic code.

The author proves the stability and the uniform convergence of a collocation method for solving Cauchy singular integral equations with regular pertubation kernel. Error estimates in the uniform norm are also given.